Wrong bone placement on object and pose mode during the eyes rigging
This is likely due to a scaled Parent bone. 

Like objects, each bone may have a parent bone that it follows by default. This parent is indicated in the Relations panel of the Bone Properties:



As noted in the properties, the default behavior for Child bones is to inherit the Rotation and Scale of the parent. The effect of this, as you've seen, is for some bones to seemingly move in Pose Mode from their original location in Edit Mode. 

To fix this simply select all bones in Pose Mode and press ALT+R and ALT+S to reset their scale and rotation.

You can easily find the parent of any bone, if there is one, by selecting a bone in Edit Mode and then Select &gt; Parent:



Anytime you extrude a bone the newly extruded bone is automatically parented, and connected to the bone it was extruded from.